Construction of Hilbert and Quot Schemes, and Its Application to the Construction of Picard Schemes (And Also a Sketch of Formal Schemes and Some Quotient Techniques)

Home , Hilbert scheme, Scheme (mathematics)

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In a sequence of talks in the Bourbaki seminar, collected under the title ‘Fondements de la Géométrie Algébriques’ (see [FGA]), he gave a sketch of the theory of descent, the construction of Hilbert and Quot schemes, and its application to the construction of Picard schemes (and also a sketch of formal schemes and some quotient techniques). The following notes give an expository account of the construction of Hilbert and Quot schemes. We assume that the reader is familiar with the basics of the language of schemes and cohomology, say at the level of chapters 2 and 3 of Hartshorne’s ‘Al- gebraic Geometry’ [H]. Some more advanced facts about flat morphisms (including the local criterion for flatness) that we need are available in Altman and Kleiman’s ‘Introduction to Grothendieck Duality Theory’ [A-K 1]. The lecture course by Vis- toli [V] on the theory of descent in this summer school contains in particular the background we need on descent. Certain advanced techniques of projective geometry, namely Castelnuovo-Mumford regularity and flattening stratification (to each of which we devote one lecture) are nicely given in Mumford’s ‘Lectures on Curves on an Algebraic Surface’ [M]. The book ‘Neron Models’ by Bosch, Lütkebohmert, Raynaud [B-L-R] contains a quick exposition of descent, quot schemes, and Picard schemes. The reader of these lecture notes is strongly urged to read Grothendieck’s original presentation in [FGA].

1 The Hilbert and Quot Functors

The Functors HilbPn The main problem addressed in this series of lectures, in its simplest form, is as follows. If S is a locally noetherian scheme, a family of subschemes of Pn n n parametrised by S will mean a closed subscheme Y ⊂ PS = PZ × S such that Y is ﬂat over S. If f : T → S is any morphism of locally noetherian schemes, then by ∗ −1 n pull-back we get a family f (Y ) = (id ×f) (Y ) ⊂ PT parametrised by T , from a family Y parametrised by S. This deﬁnes a contravariant functor HilbPn from the category of all locally noetherian schemes to the category of sets, which associates to any S the set of all such families

n HilbPn (S)= {Y ⊂ PS | Y is ﬂat over S}

Question: Is the functor HilbPn representable? Grothendieck proved that this question has an aﬃrmative answer, that is, there n exists a locally noetherian scheme HilbPn together with a family Z ⊂ PZ × HilbPn parametrised by HilbPn , such that any family Y over S is obtained as the pull-back of Z by a uniquely determined morphism ϕY : S → HilbPn . In other words, HilbPn is isomorphic to the functor Mor(−, HilbPn ).

2 r The Functors Quot⊕ OPn A family Y of subschemes of Pn parametrised by S is the same as a coherent quotient n n P sheaf q : OPS → OY on S, such that OY is ﬂat over S. This way of looking at the functor HilbPn has the following fruitful generalisation. r Let r be any positive integer. A family of quotients of ⊕ OPn parametrised by a locally noetherian scheme S will mean a pair (F, q) consisting of n (i) a coherent sheaf F on PS which is ﬂat over S, and

n r n (ii) a surjective OPS -linear homomorphism of sheaves q : ⊕ OPS →F. Two such families (F, q)and (F, q) parametrised by S will be regarded as equivalent if there exists an isomorphism f : F→F ′ which takes q to q′, that is, the following diagram commutes. r q ⊕ OPn → F k ↓ f ′ r q ′ ⊕ OPn → F This is the same as the condition ker(q) = ker(q′). We will denote by hF, qi an equivalence class. If f : T → S is a morphism of locally noetherian schemes, then r n n n P P pulling back the quotient q : ⊕ OPS → F under id ×f : T → S deﬁnes a family ∗ r n ∗ f (q) : ⊕ OPT → f (F) over T , which makes sense as tensor product is right-exact and preserves ﬂatness. The operation of pulling back respects equivalence of families,

r therefore it gives rise to a contravariant functor Quot⊕ OPn from the category of all locally noetherian schemes to the category of sets, by putting

r Quot⊕ OPn (S)= { All hF, qi parametrised by S}

r It is immediate that the functor Quot⊕ OPn satisﬁes faithfully ﬂat descent. It was proved by Grothendieck that in fact the above functor is representable on the cate-

r gory of all locally noetherian schemes by a scheme Quot⊕ OPn .

The Functors HilbX/S and QuotE/X/S

n r The above functors HilbP and Quot⊕ OPn admit the following simple generalisa- tions. Let S be a noetherian scheme and let X → S be a ﬁnite type scheme over it. Let E be a coherent sheaf on X. Let SchS denote the category of all locally noetherian schemes over S. For any T → S in SchS, a family of quotients of E parametrised by T will mean a pair (F, q) consisting of

(i) a coherent sheaf F on XT = X ×S T such that the schematic support of F is proper over T and F is ﬂat over T , together with

(ii) a surjective OXT -linear homomorphism of sheaves q : ET →F where ET is the pull-back of E under the projection XT → X. Two such families (F, q)and (F, q) parametrised by T will be regarded as equivalent if ker(q) = ker(q′)), and hF, qi will denote an equivalence class. Then as properness and ﬂatness are preserved by base-change, and as tensor-product is right exact, the

3 pull-back of hF, qi under an S-morphism T ′ → T is well-deﬁned, which gives a set-valued contravariant functor QuotE/X/S : SchS → Sets under which T 7→ { All hF, qi parametrised by T }

When E = OX , the functor QuotOX /X/S : SchS → Sets associates to T the set of all closed subschemes Y ⊂ XT that are proper and ﬂat over T . We denote this functor by HilbX/S. Note in particular that we have

n n r r n HilbP = HilbP / Spec Z and Quot⊕ OPn = Quot⊕ OPn /P / Spec Z Z Z Z

It is clear that the functors QuotE/X/S and HilbX/S satisfy faithfully ﬂat descent, so it makes sense to pose the question of their representability.

Stratification by Hilbert Polynomials Let X be a finite type scheme over a field k, together with a line bundle L. Recall that if F is a coherent sheaf on X whose support is proper over k, then the Hilbert polynomial Φ ∈ Q[λ] of F is defined by the function n i i ⊗m Φ(m)= χ(F (m)) = (−1) dimk H (X, F ⊗ L ) Xi=0 where the dimensions of the cohomologies are finite because of the coherence and properness conditions. The fact that χ(F (m)) is indeed a polynomial in m under the above assumption is a special case of what is known as Snapper’s Lemma (see Kleiman [K] for a proof). Let X → S be a finite type morphism of noetherian schemes, and let L be a line bundle on X. Let F be any coherent sheaf on X whose schematic support is proper over S. Then for each s ∈ S, we get a polynomial Φs ∈ Q[λ] which is the Hilbert polynomial of the restriction Fs = F |Xs of F to the fiber Xs over s, calculated with respect to the line bundle Ls = L|Xs . If F is flat over S then the function s 7→ Φs from the set of points of S to the polynomial ring Q[λ] is known to be locally constant on S.

This shows that the functor QuotE/X/S naturally decomposes as a co-product

Φ,L QuotE/X/S = QuotE/X/S Φ∈aQ[λ]

Φ,L where for any polynomial Φ ∈ Q[λ], the functor QuotE/X/S associates to any T the set of all equivalence classes of families hF, qi such that at each t ∈ T the Hilbert polynomial of the restriction Ft, calculated using the pull-back of L, is Φ. Correspondingly, the representing scheme QuotE/X/S, when it exists, naturally decomposes as a co-product

Φ,L QuotE/X/S = QuotE/X/S Φ∈aQ[λ]

4 Note We will generally take X to be (quasi-)projective over S, and L to be a relatively very ample line bundle. Then indeed the Hilbert and Quot functors are representable by schemes, but not in general.

Elementary Examples, Exercises

n n (1) PZ as a Quot scheme Show that the scheme PZ = Proj Z[x0,...,xn] represents the functor ϕ from schemes to sets, which associates to any S the set of all n+1 equivalence classes hF, qi of quotients q : ⊕ OS →F, where F is an invertible OS - module. As coherent sheaves on S which are OS-flat with each fiber 1-dimensional are exactly the locally free sheaves on S of rank 1, it follows that ϕ is the functor 1,OZ Quot n+1 (where in some places we write just Z for Spec Z for simplicity). ⊕ OZ/Z/Z 1,OZ n This shows that Quot n+1 = P . Under this identification, show that the ⊕ OZ/Z/Z Z 1,OZ n+1 universal family on Quot n+1 is the tautological quotient ⊕ OPn → OPn (1) ⊕ OZ/Z/Z Z Z More generally, show that if E is a locally free sheaf on a noetherian scheme S, the 1,OS P functor QuotE/S/S is represented by the S-scheme (E)= Proj SymOS E, with the ∗ tautological quotient π (E) → OP(E)(1) as the universal family. (2) Grassmannian as a Quot scheme For any integers r ≥ d ≥ 1, an explicit construction the Grassmannian scheme Grass(r, d) over Z, together with the r tautological quotient u : ⊕ OGrass(r,d) → U where U is a rank d locally free sheaf on Grass(r, d), has been given at the end of this section. A proof of the properness of π : Grass(r, d) → Spec Z is given there, together with a closed embedding m r .Grass(r, d) ֒→ P(π∗ det U)= PZ where m = d − 1 r Show that Grass(r, d) together with the quotient u : ⊕ OGrass(r,d) → U represents the contravariant functor

d,OZ Grass(r, d)= Quot r ⊕ OZ/Z/Z from schemes to sets, which associates to any T the set of all equivalence classes r hF, qi of quotients q : ⊕ OT → F where F is a locally free sheaf on T of rank d. d,OZ Therefore, Quot r exists, and equals Grass(r, d). ⊕ OZ/Z/Z Grassmannian of a vector bundle Show that for any ring A, the action of r the group GLr(A) on the free module ⊕ A induces an action of GLr(A) on the set Grass(r, d)(A), such that for any ring homomorphism A → B, the set-map Grass(r, d)(A) → Grass(r, d)(B) is equivariant with respect to the group homomorphism GLr(A) → GLr(B). (In schematic terms, this means we have an action of the group-scheme GLr,Z on Grass(r, d).) Using the above show that, more generally, if S is a scheme and E is a locally d,OS free OS-module of rank r, the functor Grass(E,d) = QuotE/S/S on all S-schemes which by definition associates to any T the set of all equivalence classes hF, qi of quotients q : ET →F where F is a locally free sheaf on T of rank d, is representable. The representing scheme is denoted by Grass(E,d) and is called the rank d relative Grassmannian of E over S. It parametrises a universal quotient π∗E → F where π : Grass(E,d) → S is the projection. Show that the determinant line bundle d F V 5 on Grass(E,d) is relatively very ample over S, and it gives a closed embedding d d Grass(E,d) ֒→ P(π∗ F) ⊂ P( E). (The properness of the embedding follows from the propernessV of π : Grass(VE,d) → S, which follows locally over S by base- change from properness of Grass(r, d) over Z – see Exercise (5) or (7) below.) Grassmannian of a coherent sheaf If E is a coherent sheaf on S, not neces- d,OS sarily locally free, then by definition the functor Grass(E,d) = QuotE/S/S on all S-schemes associates to any T the set of all equivalence classes hF, qi of quotients ′ q : ET → F where F is a locally free on T of rank d. If r : E → E is a surjection of coherent sheaves on S, then show that the induced morphism of functors Grass(E,d) → Grass(E′,d), which sends hF, qi 7→ hF, q◦ri, is a closed embedding. From this, by locally expressing a coherent sheaf as a quotient of a vector bundle, show that Grass(E,d) is representable even when E is a coherent sheaf on S which is not necessarily locally free. The representing scheme Grass(E,d) is proper over S, as locally over S it is a closed subscheme of the Grassmannian of a vector bundle. Show by arguing locally over S that the line bundle d F on Grass(E,d) is relatively very ample over S, and therefore by using propernessV conclude that Grass(E,d) is projective over S. (3) Grassmannian as a Hilbert scheme Let Φ=1 ∈ Q[λ]. Then the Hilbert 1,O(1) Pn r+λ Q scheme HilbPn is Z itself. More generally, let Φr = r ∈ [λ] where r ≥ 0. Φr,O(1) The Hilbert scheme HilbPn is isomorphic to the Grassmannian scheme Grass(n+ 1,r + 1) over Z. This can be seen via the following steps, whose detailed verification is left to the reader as an exercise. (i) The Grassmannian scheme Grass(n +1,r + 1) over Z parametrises a tautological n family of subschemes of P with Hilbert polynomial Φr. Therefore we get a natural Φr,O(1) transformation hGrass(n+1,r+1) → HilbPn . n (ii) Any closed subscheme Y ⊂ Pk with Hilbert polynomial Φr, where k is any field, r n is isomorphic to Pk embedded linearly in Pk over k. If V is a vector bundle over a noetherian base S, and if Y ⊂ P(V ) is a closed subscheme flat over S with each schematic fiber Ys an r-dimensional linear subspace of the projective space P(Vs), then Y is defines a rank r + 1 quotient vector bundle V = π∗OP(V )(1) → π∗OY (1) where π : P(V ) → S denotes the projection. This gives a natural transformation Φr,O(1) HilbPn → hGrass(n+1,r+1). (iii) The above two natural transformations are inverses of each other.

n n+λ n−d+λ (4) Hilbert scheme of hypersurfaces in P Let Φd = n − n ∈ Q[λ] Φd,O(1) m where d ≥ 1. The Hilbert scheme HilbPn is isomorphic to PZ where m = n+d d − 1. This can be seen from the following steps, which are left as exercises. n (i) Any closed subscheme Y ⊂ Pk with Hilbert polynomial Φd, where k is any ﬁeld, n n is a hypersurface of degree d in Pk . Hint: If Y ⊂ Pk is a closed subscheme with Hilbert polynomial of degree n − 1, then show that the schematic closure Z of the n hight 1 primary components is a hypersurface in Pk with deg(Z) = deg(Y ). n n (ii) Any family Y ⊂ PS is a Cartier divisor in PS. It gives rise to a line subbundle m n n P π∗(IY ⊗ OPS (d)) ⊂ π∗OPS (d), which deﬁnes a natural morphism fY : S → Z where

6 n+d Φd m m = d − 1. This gives a morphism of functors HilbPn → P where we denote m m P hPZ simply by . m (iii) The scheme PZ parametrises a tautological family of hypersurfaces of degree d, m Φd,O(1) which gives a morphism of functors P → HilbPn in the reverse direction. These are inverses of each other. (5) Base-change property of Hilbert and Quot schemes Let S be a noetherian scheme, X a ﬁnite-type scheme over S, and E a coherent sheaf on X. If T → S is a morphism of noetherian schemes, then show that there is a natural isomorphism of functors QuotET /XT /T → QuotE/X/S ×hS hT . Consequently, if QuotE/X/S exists, then so does QuotET /XT /T , which is naturally isomorphic to QuotE/X/S ×ST . Φ,L One can prove a similar statement involving QuotE/X/S . In particular, HilbX/S and Φ,L HilbX/S , when they exist, base-change correctly.

(6) Descent condition in the fpqc topology If U is an S-scheme and (fi : Ui → U) is an open cover of U in the fpqc topology, then show that the following sequence of sets is exact:

→ QuotE/X/S(U) → QuotE/X/S(Ui) → QuotE/X/S (Ui ×U Uj) Yi Yi,j

(7) Valuative criterion for properness When X → S is proper, show that the morphism of functors QuotE/X/S → hS satisfies the valuative criterion of properness with respect to discrete valuation rings, that is, if R is a discrete valuation ring together with a given morphism Spec R → S making it an S-scheme, show that the restriction map QuotE/X/S(Spec R) → QuotE/X/S(Spec K) is bijective, where K is the quotient field of R and Spec K is regarded as an S-scheme in the obvious way. (8) Counterexample of Hironaka Hironaka constructed a 3-dimensional smooth proper scheme X over complex numbers C, together with a free action of the group G = Z/(2), for which the quotient X/G does not exist as a scheme. (See Example 3.4.1 in Hartshorne [H] Appendix B for construction of X. We leave the definition of the G action and the proof that X/G does not exist to the reader.) In particular, this means the Hilbert functor HilbX/C is not representable by a scheme.

Construction of Grassmannian The following explicit construction of the Grassmannian scheme Grass(r, d) over Z is best understood as the construction of a quotient GLd,Z\V , where V is the scheme of all d × r matrices of rank d, and the group-scheme GLd,Z acts on V on the left by matrix multiplication. However, we will not use the language of group-scheme actions here, instead, we give a direct elementary construction of the Grassmannian scheme. The reader can take d = 1 in what follows, in a ﬁrst reading, to get the special case r−1 Grass(r, 1) = PZ , which has another construction as Proj Z[x1,...,xr].

7 Construction by gluing together affine patches For any integers r ≥ d ≥ 1, the Grassmannian scheme Grass(r, d) over Z, together with the tautological quotient r u : ⊕ OGrass(r,d) → U where U is a rank d locally free sheaf on Grass(r, d), can be explicitly constructed as follows. If M is a d × r-matrix, and I ⊂{1,...,r} with cardinality #(I) equal to d, the I th minor MI of M will mean the d × d minor of M whose columns are indexed by I. For any subset I ⊂ {1,...,r} with #(I) = d, consider the d × r matrix XI whose I I I the minor XI is the d × d identity matrix 1d×d, while the remaining entries of X I I are independent variables xp,q over Z. Let Z[X ] denote the polynomial ring in the I I I variables xp,q, and let U = Spec Z[X ], which is non-canonically isomorphic to the d(r−d) affine space AZ . I I I I For any J ⊂{1,...,r} with #(J) = d, let PJ = det(XJ ) ∈ Z[X ] where XJ is the I I I I I I J th minor of X . Let UJ = Spec Z[X , 1/PJ ] the open subscheme of U where PJ I I −1 I is invertible. This means the d × d-matrix XJ admits an inverse (XJ ) on UJ . J J I I For any I and J, a ring homomorphism θI,J : Z[X , 1/PI ] → Z[X , 1/PJ ] is defined J as follows. The images of the variables xp,q are given by the entries of the matrix J I −1 I J I formula θI,J (X )=(XJ ) X . In particular, we have θI,J (PI )=1/PJ , so the map J J extends to Z[X , 1/PI ]. I I Note that θI,I is identity on UI = U , and we leave it to the reader to verify that for any three subsets I, J and K of {1,...,r} of cardinality d, the co-cycle condition I r θI,K = θI,J θJ,K is satisfied. Therefore the schemes U , as I varies over all the d different subsets of {1,...,r} of cardinality d, can be glued together by the co-cycle I (θI,J ) to form a finite-type scheme Grass(r, d) over Z. As each U is isomorphic to d(r−d) AZ , it follows that Grass(r, d) → Spec Z is smooth of relative dimension d(r−d). Separatedness The intersection of the diagonal of Grass(r, d) with U I × U J can I J be seen to be the closed subscheme ∆I,J ⊂ U × U defined by entries of the matrix J I J formula XI X − X = 0, and so Grass(r, d) is a separated scheme. Properness We now show that π : Grass(r, d) → Spec Z is proper. It is enough to verify the valuative criterion of properness for discrete valuation rings. Let R be a dvr, K its quotient field, and let ϕ : Spec K → Grass(r, d) be a morphism. This is given by a ring homomorphism f : Z[XI ] → K for some I. Having fixed one such I I, next choose J such that ν(f(PJ )) is minimum, where ν : K → Z {∞} denotes I I I the discrete valuation. As PI = 1, note that ν(f(PJ )) ≤ 0, thereforeSf(PJ ) =6 0 in I K and so the matrix f(XJ ) lies in GLd(K). Now consider the homomorphism g : Z[XJ ] → K defined by entries of the matrix formula J I −1 I g(X )= f((XJ ) X ) Then g defines the same morphism ϕ : Spec K → Grass(r, d), and moreover all J J J d × d minors XK satisfy ν(g(PK)) ≥ 0. As the minor XJ is identity, it follows J J from the above that in fact ν(g(xp,q)) ≥ 0 for all entries of X . Therefore, the map g : Z[XJ ] → K factors uniquely via R⊂K. The resulting morphism of schemes Spec R → U J ֒→ Grass(r, d) prolongs ϕ : Spec K → Grass(r, d). We have already

8 checked separatedness of Grass(r, d), so now we see that Grass(r, d) → Spec Z is proper. Universal quotient We next define a rank d locally free sheaf U on Grass(r, d) r I together with a surjective homomorphism ⊕ OGrass(r,d) → U. On each U we define I r d I a surjective homomorphism u : ⊕ OU I → ⊕ OU I by the matrix X . Compatible I with the co-cycle (θI,J ) for gluing the affine pieces U , we give gluing data (gI,J ) for d gluing together the trivial bundles ⊕ OU I by putting

I −1 I gI,J =(XJ ) ∈ GLd(UJ ) This is compatible with the homomorphisms uI, so we get a surjective homomor- r phism u : ⊕ OGrass(r,d) → U.

Projective embedding As U is given by the transition functions gI,J described above, the determinant line bundle det(U) is given by the transition functions I I det(gI,J )=1/PJ ∈ GL1(UJ ). For each I, we deﬁne a global section

σI ∈ Γ(Grass(r, d), det(U))

J J by putting σI |U J = PI ∈ Γ(U , OU J ) in terms of the trivialization over the open J cover (U ). We leave it to the reader to verify that the sections σI form a linear system which is base point free and separates points relative to Spec Z, and so gives m r an embedding of Grass(r, d) into PZ where m = d − 1. This is a closed embedding by the properness of π : Grass(r, d) → Spec Z. In particular, det(U) is a relatively very ample line bundle on Grass(r, d) over Z.

Note The σI are known as the Plücker coordinates, and these satisfy certain quadratic polynomials known as the Plücker relations, which define the projective image of the Grassmannian. We will not need these facts.

2 Castelnuovo-Mumford Regularity

Mumford’s deployment of m-regularity led to a simpliﬁcation in the construction of Quot schemes. The original construction of Grothendieck had instead relied on Chow coordinates. Let k be a ﬁeld and let F be a coherent sheaf on the projective space Pn over k. Let m be an integer. The sheaf F is said to be m-regular if we have

Hi(Pn, F(m − i)) = 0 for each i ≥ 1.

The deﬁnition, which may look strange at ﬁrst sight, is suitable for making inductive arguments on n = dim(Pn) by restriction to a suitable hyperplane. If H ⊂ Pn is a hyperplane which does not contain any associated point of F, then we have a short exact sheaf sequence

α 0 →F(m − i − 1) →F(m − i) →FH (m − i) → 0

9 where the map α is locally given by multiplication with a deﬁning equation of H, hence is injective. The resulting long exact cohomology sequence

i n i n i+1 n . . . → H (P , F(m − i)) → H (P , FH(m − i)) → H (P , F(m − i − 1)) → . . . shows that if F is m-regular, then so is its restriction FH (with the same value for m) to a hyperplane H ≃ Pn−1 which does not contain any associated point of F. Note that whenever F is coherent, the set of associated points of F is finite, so there will exist at least one such hyperplane H when the field k is infinite. The following lemma is due to Castelnuovo, according to Mumford’s Curves on a surface.

Lemma 2.1 If F is an m-regular sheaf on Pn then the following statements hold: 0 n 0 n 0 n (a) The canonical map H (P , OPn (1)) ⊗ H (P , F(r)) → H (P , F(r + 1)) is surjective whenever r ≥ m. (b) We have Hi(Pn, F(r))=0 whenever i ≥ 1 and r ≥ m − i. In other words, if F is m-regular, then it is m′-regular for all m′ ≥ m. (c) The sheaf F(r) is generated by its global sections, and all its higher cohomologies vanish, whenever r ≥ m.

Proof As the cohomologies base-change correctly under a field extension, we can assume that the field k is infinite. We argue by induction on n. The statements (a), (b) and (c) clearly hold when n = 0, so next let n ≥ 1. As k is infinite, there exists a hyperplane H which does not contain any associated point of F, so that the n−1 restriction FH is again m-regular as explained above. As H is isomorphic to Pk , by the inductive hypothesis the assertions of the lemma hold for the sheaf FH . When r = m − i, the equality Hi(Pn, F(r)) = 0 in statement (b) follows for all n ≥ 0 by definition of m-regularity. To prove (b), we now proceed by induction on r where r ≥ m − i + 1. Consider the exact sequence

i n i n i H (P , F(r − 1)) → H (P , F(r)) → H (H, FH (r))

By inductive hypothesis for r−1 the ﬁrst term is zero, while by inductive hypothesis for n − 1 the last term is zero, which shows that the middle term is zero, completing the proof of (b). Now consider the commutative diagram

0 n 0 n σ 0 0 H (P , F(r)) ⊗ H (P , OPn (1)) → H (H, FH (r)) ⊗ H (H, OH (1)) ↓ µ ↓ τ 0 n α 0 n νr+1 0 H (P , F(r)) → H (P , F(r + 1)) → H (H, FH (r + 1)) The top map σ is surjective, for the following reason: By m-regularity of F and using the statement (b) already proved, we see that H1(Pn, F(r−1)) = 0 for r ≥ m, 0 n 0 and so the restriction map νr : H (P , F(r)) → H (H, FH (r)) is surjective. Also, 0 n 0 the restriction map ρ : H (P , OPn (1)) → H (H, OH (1)) is surjective. Therefore the tensor product σ = νr ⊗ ρ of these two maps is surjective.

10 The second vertical map τ is surjective by inductive hypothesis for n − 1 = dim(H).

Therefore, the composite τ ◦ σ is surjective, so the composite νr+1 ◦ µ is surjective, 0 n hence H (P , F(r +1)) = im(µ) + ker(νr+1). As the bottom row is exact, we get H0(Pn, F(r +1)) = im(µ) + im(α). However, we have im(α) ⊂ im(µ), as the map α is given by tensoring with a certain section of OPn (1) (which has divisor H). Therefore, H0(Pn, F(r +1)) = im(µ). This completes the proof of (a) for all n. 0 n 0 n 0 n To prove (c), consider the map H (P , F(r)) ⊗ H (P , OPn (p)) → H (P , F(r + p)), which is surjective for r ≥ m and p ≥ 0 as follows from a repeated use of (a). For p ≫ 0, we know that H0(Pn, F(r + p)) is generated by its global sections. It follows that H0(Pn, F(r)) is also generated by its global sections for r ≥ m. We already know from (b) that Hi(Pn, F(r)) = 0 for i ≥ 1 when r ≥ m. This proves (c), completing the proof of the lemma.

Remark 2.2 The following fact, based on the diagram used in the course of the above proof, will be useful later: With notation as above, let the restriction map 0 n 0 νr : H (P , F(r)) → H (H, FH (r)) be surjective. Also, let FH be r-regular, so that 0 0 0 by Lemma 2.1.(a) the map H (H, OH (1)) ⊗ H (H, FH(r)) → H (H, FH (r + 1)) is 0 n 0 surjective. Then the restriction map νr+1 : H (P , F(r + 1)) → H (H, FH (r + 1)) is again surjective. As a consequence, if FH is m regular and if for some r ≥ m the 0 n 0 restriction map νr : H (P , F(r)) → H (H, FH (r)) is surjective, then the restriction 0 n 0 map νp : H (P , F(p)) → H (H, FH (p)) is surjective for all p ≥ r.

Exercise Find all the values of m for which the invertible sheaf OPn (r) is m-regular. Exercise Suppose 0 → F ′ →F →F ′′ → 0 is an exact sequence of coherent sheaves on Pn. Show that if F ′ and F ′′ are m-regular, then F is also m-regular, if F ′ is (m+1)-regular and F is m-regular, then F ′′ is m-regular, and if F is m-regular and F ′′ is (m − 1)-regular, then F ′ is m-regular.

The use of m-regularity for making Quot schemes is via the following theorem.

Theorem 2.3 (Mumford) For any non-negative integers p and n, there exists a polynomial Fp,n in n +1 variables with integral coefficients, which has the following property: Let k be any field, and let Pn denote the n-dimensional projective space over k. Let n p F be any coherent sheaf on P , which is isomorphic to a subsheaf of ⊕ OPn . Let the Hilbert polynomial of F be written in terms of binomial coefficients as

n r χ(F(r)) = a i i Xi=0 where a0,...,an ∈ Z.

Then F is m-regular, where m = Fp,n(a0,...,an).

11 Proof (Following Mumford [M]) As before, we can assume that k is infinite. We argue by induction on n. When n = 0, clearly we can take Fp,0 to be any polynomial. Next, let n ≥ 1. Let H ⊂ Pn be a hyperplane which does not contain any of the p finitely many associated points of ⊕ OPn /F (such an H exists as k is infinite). Then the following torsion sheaf vanishes:

1 p n TorOPn (OH , ⊕ OP /F)=0

p p Therefore the sequence 0 →F→⊕ OPn → ⊕ OPn /F → 0 restricts to H to give p p a short exact sequence 0 → FH → ⊕ OH → ⊕ OH /FH → 0. This shows that FH p n−1 is isomorphic to a subsheaf of ⊕ O n−1 (under an identification of H with P ), Pk k which is a basic step needed for our inductive argument. Note that F is torsion free if non-zero, and so we have a short exact sequence 0 → F(−1) →F→FH → 0. From the associated cohomology sequence we get n r n r−1 n r−1 χ(FH (r)) = χ(F(r)) − χ(F(r − 1)) = i=0 ai i − i=0 ai i = i=0 ai i−1 = n−1 r j=0 bj j where the coefficients b0,...,bP n−1 have expressionsP bj =Pgj(a0,...,an) whereP the gj are polynomials with integral coefficients independent of the field k and the sheaf F. (Exercise: Write down the gj explicitly.)

By inductive hypothesis on n − 1 there exists a polynomial Fp,n−1(x0,...,xn−1) such that FH is m0-regular where m0 = Fp,n−1(b0,...,bn−1). Substituting bj = gj(a0,...,an), we get m0 = G(a0,...,an), where G is a polynomial with integral coeﬃcients independent of the ﬁeld k and the sheaf F. For m ≥ m0 − 1, we therefore get a long exact cohomology sequence

0 0 νm 0 1 1 0 → H (F(m − 1)) → H (F(m)) → H (FH (m)) → H (F(m − 1)) → H (F(m)) → 0 → . . .

which for i ≥ 2 gives isomorphisms Hi(F(m − 1)) →∼ Hi(F(m)). As we have Hi(F(m)) =0 for m ≫ 0, these equalities show that

i H (F(m)) = 0 for all i ≥ 2 and m ≥ m0 − 2.

The surjections H1(F(m − 1)) → H1(F(m)) show that the function h1(F(m)) is a monotonically decreasing function of the variable m for m ≥ m0 − 2. We will in 1 fact show that for m ≥ m0, the function h (F(m)) is strictly decreasing till its value reaches zero, which would imply that

1 1 H (F(m)) =0 for m ≥ m0 + h (F(m0)).

1 Next we will put a suitable upper bound on h (F(m0)) to complete the proof of 1 1 the theorem. Note that h (F(m − 1)) ≥ h (F(m)) for m ≥ m0, and moreover 0 equality holds for some m ≥ m0 if and only if the restriction map νm : H (F(m)) → 0 H (FH(m)) is surjective. As FH is m-regular, it follows from Remark 2.2 that 0 0 the restriction map νj : H (F(j)) → H (FH (j)) is surjective for all j ≥ m, so h1(F(j − 1)) = h1(F(j)) for all j ≥ m. As h1(F(j))= 0 for j ≫ 0, this establishes 1 our claim that h (F(m)) is strictly decreasing for m ≥ m0 till its value reaches zero.

12 1 p To put a bound on h (F(m0)), we use the fact that as F ⊂ ⊕ OPn we must 0 0 n+r n have h (F(r)) ≤ ph (OP (r)) = p n . From the already established fact that i h (F(m)) = 0 for all i ≥ 2 and m ≥m0 − 2, we now get

1 0 h (F(m0)) = h (F(m0)) − χ(F(m0)) n + m n m ≤ p 0 − a 0 n i i Xi=0 = P (a0,...an) where P (a0,...,an) is a polynomial expression in a0,...,an, obtained by substituting m0 = G(a0,...,an) in the second line of the above (in)equalities. Therefore, the coeﬃcients of the corresponding polynomial P (x0,...,xn) are again independent of 1 the ﬁeld k and the sheaf F. Note moreover that as h (F(m0)) ≥ 0, we must have P (a0,...,an) ≥ 0. Substituting in an earlier expression, we get

1 H (F(m)) =0 for m ≥ G(a0,...,an)+ P (a0,...,an)

Taking Fp,n(x0,...,xn) to be G(x0,...,xn) + P (x0,...,xn), and noting the fact that P (a0,...,an) ≥ 0, we see that F is Fp,n(a0,...,an)-regular. This completes the proof of the theorem.

Exercise Write down such polynomials Fp,n.

3 Semi-Continuity and Base-Change Base-change without Flatness The following lemma on base-change does not need any ﬂatness hypothesis. The price paid is that the integer r0 may depend on φ.

Lemma 3.1 Let φ : T → S be a morphism of noetherian schemes, let F a coherent n sheaf on PS, and let FT denote the pull-back of F under the induced morphism n n n n PT → PS. Let πS : PS → S and πT : PT → T denote the projections. Then there exists an integer r0 such that the base-change homomorphism

∗ φ πS∗ F(r) → πT ∗ FT (r) is an isomorphism for all r ≥ r0.

Proof As base-change holds for open embeddings, using a finite affine open cover −1 Ui of S and a finite affine open cover Vi,j of each φ (Ui) (which is possible by noetherian hypothesis), it is enough to consider the case where S and T are affine.

13 Note that for all integers i, the base-change homomorphism

∗ n n φ πS∗OPS (i) → πT ∗OPT (i)

n n is an isomorphism. Moreover, if a and b are any integers and if f : OPS (a) → OPS (b) n n n P is any homomorphism and fT : OPT (a) → OPT (b) denotes its pull-back to T , then for all i we have the following commutative diagram where the vertical maps are base-change isomorphisms.

∗ φ πS∗f(i) ∗ n ∗ n φ πS∗OPS (a + i) → φ πS∗OPS (b + i) ↓ ↓ πT ∗fT (i) n n πT ∗OPT (a + i) → πT ∗OPT (b + i)

As S is noetherian and aﬃne, there exists an exact sequence

p u q v n n ⊕ OPS (a) → ⊕ OPS (b) →F → 0

n for some integers a, b, p ≥ 0, q ≥ 0. Its pull-back to PT is an exact sequence

p uT q vT n n ⊕ OPT (a) → ⊕ OPS (b) → FT → 0

Let G = ker(v) and let H = ker(vT ). For any integer r, we get exact sequences

p q 1 n n πS∗ ⊕ OPS (a + r) → πS∗ ⊕ OPS (b + r) → πS∗F(r) → R πS∗G(r) and

p q 1 n n πT ∗ ⊕ OPT (a + r) → πT ∗ ⊕ OPT (b + r) → πT ∗FT (r) → R πT ∗H(r)

1 1 There exists an integer r0 such that R πS ∗G(r) = 0 and R πT ∗H(r) = 0 for all r ≥ r0. Hence for all r ≥ r0, we have exact sequences

p πS∗u(r) q πS∗v(r) n n πS∗ ⊕ OPS (a + r) → ⊕ OPS (b + r) → πS∗F(r) → 0 and

p πT ∗uT (r) q πT ∗vT (r) n n πT ∗ ⊕ OPT (a + r) → πT ∗ ⊕ OPT (b + r) → πT ∗FT (r) → 0

Pulling back the second-last exact sequence under φ : T → S, we get the commutative diagram with exact rows

∗ ∗ φ πS∗u φ πS∗v ∗ p n ∗ q n ∗ φ πS∗ ⊕ OPS (a + r) → φ ⊕ OPS (b + r) → φ πS∗F(r) → 0 ↓ ↓ ↓ πT ∗uT (r) πT ∗vT (r) p n q n πT ∗ ⊕ OPT (a + r) → πT ∗ ⊕ OPT (b + r) → πT ∗FT (r) → 0 in which the ﬁrst row is exact by the right-exactness of tensor product. The vertical maps are base-change homomorphisms, the ﬁrst two of which are isomorphisms for

14 ∗ all r. Therefore by the ﬁve lemma, φ πS∗F(r) → πT ∗FT (r) is an isomorphism for all r ≥ r0.

The following elementary proof of the above result is taken from Mumford [M]: Let M be the ∼ ∗ graded OS-module ⊕m∈Z πS∗F(m), so that F = M . Let φ M be the graded OT -module which ∗ ∼ is the pull-back of M. Then we have FT = (φ M) . On the other hand, let N = ⊕m∈Z πT ∗FT (m), ∼ so that we have FT = N . Therefore, in the category of graded OT [x0,...,xn]-modules, we get an induced equivalence between φ∗M and N, which means the natural homomorphisms of graded ∗ pieces (φ M)m → Nm are isomorphisms for all m ≫ 0.

Flatness of F from Local Freeness of π∗F(r)

n Lemma 3.2 Let S be a noetherian scheme and let F be a coherent sheaf on PS. Suppose that there exists some integer N such that for all r ≥ N the direct image π∗F(r) is locally free. Then F is ﬂat over S.

Proof Consider the graded module M = ⊕r≥N Mr over OS, where Mr = π∗F(r). ∼ n The sheaf F is isomorphic to the sheaf M on PS = ProjS OS[x0,...,xn] made from the graded sheaf M of OS-modules. As each Mr is ﬂat over OS, so is M.

Therefore for any xi the localisation Mxi is flat over OS. There is a grading on q Mxi , indexed by Z, defined by putting deg(vp/xi ) = p − q for vp ∈ Mp (this is well-defined). Hence the component (Mxi )0 of degree zero, being a direct summand ∼ of Mxi , is again flat over OS. But by definition of M , this is just Γ(Ui, F), where n n Ui = SpecS OS [x0/xi,...,xn/xi] ⊂ PS. As the Ui form an open cover of PS, it follows that F is flat over OS.

Exercise Show that the converse of the above lemma holds: if F is ﬂat over S then π∗F(r) is locally free for all suﬃciently large r. Grothendieck Complex for Semi-Continuity The following is a very important basic result of Grothendieck, and the complex K· occurring in it is called the Grothendieck complex.

Theorem 3.3 Let π : X → S be a proper morphism of noetherian schemes where S = Spec A is affine, and let F be a coherent OX -module which is flat over OS. Then there exists a finite complex

0 → K0 → K1 → . . . → Kn → 0 of ﬁnitely generated projective A-modules, together with a functorial A-linear isomorphism p ∼ p · H (X, F ⊗A M) → H (K ⊗A M) on the category of all A-modules M.

15 The above theorem is the foundation for all results about direct images and base- change for ﬂat families of sheaves, such as Theorem 3.7. As another consequence of the above theorem, we have the following.

Theorem 3.4 ([EGA] III 7.7.6) Let S be a noetherian scheme and π : X → S a proper morphism. Let F be a coherent sheaf on X which is ﬂat over S. Then there exists a coherent sheaf Q on S together with a functorial OS-linear isomorphism

∗ θG : π∗(F ⊗OX π G) → HomOS (Q, G) on the category of all quasi-coherent sheaves G on S. By its universal property, the pair (Q, θ) is unique up to a unique isomorphism.

Proof If S = Spec A, then we can take Q to be the coherent sheaf associated to the A-module Q which is the cokernel of the transpose ∂∨ : (K1)∨ → (K0)∨ where ∂ : K0 → K1 is the diﬀerential of any chosen Grothendieck complex of A-modules 0 → K0 → K1 → . . . → Kn → 0 for the sheaf F, whose existence is given by Theorem 3.3. For any A-module M, the right-exact sequence (K1)∨ → (K0)∨ → Q → 0 with M gives on applying HomA(−, M) a left-exact sequence

0 1 0 → HomA(Q, M) → K ⊗A M → K ⊗A M

Therefore by Theorem 3.3, we have an isomorphism

A 0 θM : H (XA, FA ⊗A M) → HomA(Q, M)

Thus, the pair (Q, θA) satisﬁes the theorem when S = Spec A. More generally, we can cover S by aﬃne open subschemes. Then on their overlaps, the resulting pairs (Q, θ) glue together by their uniqueness.

A linear scheme V → S over a noetherian base scheme S is a scheme of the form

Spec SymOS Q where Q is a coherent sheaf on S. This is naturally a group scheme. Linear schemes generalise the notion of (geometric) vector bundles, which are the special case where Q is locally free of constant rank.

The zero section V0 ⊂ V of a linear scheme V = Spec SymOS Q is the closed subscheme deﬁned by the ideal generated by Q. Note that the projection V0 → S is an isomorphism, and V0 is just the image of the zero section 0 : S → V of the group-scheme.

Theorem 3.5 ([EGA] III 7.7.8, 7.7.9) Let S be a noetherian scheme and π : X → S a projective morphism. Let E and F be coherent sheaves on X. Consider the set-valued contravariant functor Hom(E, F) on S-schemes, which associates to any

T → S the set of all OXT -linear homomorphisms HomXT (ET , FT ) where ET and FT denote the pull-backs of E and F under the projection XT → X. If F is ﬂat over S, then the above functor is representable by a linear scheme V over S.

16 Proof First note that if E is a locally free OX -module, then Hom(E, F) is the 0 ∨ ∨ functor T 7→ H (XT , (F ⊗OX E )T ). The sheaf F ⊗OX E is again ﬂat over S, so we can apply Theorem 3.4 to get a coherent sheaf Q, such that we have π∗(F ⊗OX ∨ ∗ E ⊗OX π G)= HomOS (Q, G) for all quasi-coherent sheaves G on S. In particular, if f : Spec R → S is any morphism then taking G = f∗OR we get

MorS(Spec R, Spec SymOS Q) = HomOS −mod(Q, f∗OR) 0 ∨ ∗ = H (X, F ⊗OX E ⊗OX π f∗OR) 0 ∨ = H (XR, (F ⊗OX E )R)

= HomXR (ER, FR).

This shows that V = Spec SymOS Q is the required linear scheme when E is locally free on X. More generally for an arbitrary coherent E, over any aﬃne open U ⊂ S there exist vector bundles E1 and E0 on XU and a right exact sequence E1 → E0 →E → 0. (This is where we need projectivity of X → S. Instead, we could have assumed just properness together with the condition that locally over S we have such a resolution of E.) Then applying the above argument to the functors Hom(E1, F) and Hom(E0, F), we get coherent sheaves Q1 and Q0 on U, and from the natural transformation Hom(E0, F) → Hom(E1, F) induced by the homomorphism E1 → E0, we get a homomorphism Q1 → Q0. Let QU be its cokernel, and put

VU = Spec SymOU QU . It follows from its deﬁnition (and the left exactness of Hom) that the scheme VU has the desired universal property over U. Therefore all such VU , as U varies over an aﬃne open cover of S, patch together to give the desired linear scheme V. (In sheaf terms, the sheaves QU will patch together to give a coherent sheaf Q on S with V = Spec SymOS Q.)

Remark 3.6 In particular, note that the zero section V0 ⊂ V is where the universal homomorphism vanishes. If f ∈ HomXT (ET , FT ) deﬁnes a morphism −1 ′ ϕf : T → V, then the inverse image f V0 is a closed subscheme T of T with the universal property that if U → T is any morphism of schemes such that the pull-back of f is zero, then U → T factors via T ′. Base-change for Flat Sheaves The following is the main result of Grothendieck on base change for ﬂat families of sheaves, which is a consequence of Theorem 3.3.

Theorem 3.7 Let π : X → S be a proper morphism of noetherian schemes, and let F be a coherent OX -module which is ﬂat over OS. Then the following statements hold: i (1) For any integer i the function s 7→ dimκ(s) H (Xs, Fs) is upper semi-continuous on S, i i (2) The function s 7→ i(−1) dimκ(s) H (Xs, Fs) is locally constant on S. (3) If for some integerPi, there is some integer d ≥ 0 such that for all s ∈ S we have i i i−1 dimκ(s) H (Xs, Fs) = d, then R π∗F is locally free of rank d, and (R π∗F)s → i−1 H (Xs, Fs) is an isomorphism for all s ∈ S.

17 i i (4) If for some integer i and point s ∈ S the map (R π∗F)s → H (Xs, Fs) is surjective, then there exists an open subscheme U ⊂ S containing s such that for any quasi-coherent OU -module G the natural homomorphism

(Riπ F ) ⊗ G → Riπ (F ⊗ π ∗G) U ∗ XU OU U ∗ XU OXU U

−1 is an isomorphism, where XU = π (U) and πU : XU → U is induced by π. In i i ′ particular, (R π∗F)s′ → H (Xs′ , Fs′ ) is an isomorphism for all s in U. i i (5) If for some integer i and point s ∈ S the map (R π∗F)s → H (Xs, Fs) is surjective, then the following conditions (a) and (b) are equivalent: i−1 i−1 (a) The map (R π∗F)s → H (Xs, Fs) is surjective. i (b) The sheaf R π∗F is locally free in a neighbourhood of s in S.

See for example Hartshorne [H] Chapter III, Section 12 for a proof. It is possible to replace the use of the formal function theorem in [H] (or the original argument in [EGA] based on completions) in proving the statement (4) above, with an elementary argument based on applying Nakayama lemma to the Grothendieck complex.

4 Generic Flatness and Flattening Stratiﬁcation

Lemma on Generic Flatness Lemma 4.1 Let A be a noetherian domain, and B a ﬁnite type A algebra. Let M be a ﬁnite B-module. Then there exists an f ∈ A, f =06 , such that the localisation Mf is a free module over Af .

Proof Over the quotient field K of A, the K-algebra BK = K ⊗A B is of finite type, and MK = K ⊗A M is a finite module over BK . Let n be the dimension of the support of MK over Spec BK . We argue by induction on n, starting with −1 n = −1 which is the case when MK = 0. In this case, as K ⊗A M = S M where S = A −{0}, each v ∈ M is annihilated by some non-zero element of A. Taking a finite generating set, and a common multiple of corresponding annihilating elements, we see there exists an f =6 0 in A with fM = 0. Hence Mf = 0, proving the lemma when n = −1. Now let n ≥ 0, and let the lemma be proved for smaller values. As B is noetherian and M is assumed to be a finite B-module, there exists a finite filtration

0= M0 ⊂ . . . ⊂ Mr = M where each Mi is a B-submodule of M such that for each i ≥ 1 the quotient module Mi/Mi−1 is isomorphic to B/pi for some prime ideal pi in B. Note that if 0 → M ′ → M → M ′′ → 0 is a short exact sequence of B-modules, ′ ′′ ′ ′′ and if f and f are non-zero elements of A such that Mf ′ and Mf ′′ are free over ′ ′′ respectively Af ′ and Af ′′ , then Mf is a free module over Af where f = f f . We

18 will use this fact repeatedly. Therefore it is enough to prove the result when M is of the form B/p for a prime ideal p in B. This reduces us to the case where B is a domain and M = B.

As by assumption K ⊗A B has dimension n ≥ 0 (that is, K ⊗A B is non-zero), the map A → B must be injective. By Noether normalisation lemma, there exist elements b1,...,bn ∈ B, such that K ⊗A B is finite over its subalgebra K[b1,...,bn] and the elements b1,...,bn are algebraically independent over K. (For simplicity of notation, we write 1 ⊗ b simply as b.) If g =6 0 in A is chosen to be a ‘common denominator’ for coefficients of equations of integral dependence satisfied by a finite set of algebra generators for K ⊗A B over K[b1,...,bn], we see that Bg is finite over Ag[b1,...,bn].

Let m be the generic rank of the finite module Bg over the domain Ag[b1,...,bn]. Then we have a short exact sequence of Ag[b1,...,bn]-modules of the form ⊕m 0 → Ag[b1,...,bn] → Bg → T → 0 where T is a finite torsion module over Ag[b1,...,bn]. Therefore, the dimension of the support of K ⊗Ag T as a K ⊗Ag (Bg)-module is strictly less than n. Hence by induction on n (applied to the data Ag, Bg, T ), there exists some h =6 0 in A with Th free over Agh. Taking f = gh, the lemma follows from the above short exact sequence. The above theorem has the following consequence, which follows by restricting at- tention to a non-empty affine open subscheme of S. Theorem 4.2 Let S be a noetherian and integral scheme. Let p : X → S be a finite type morphism, and let F be a coherent sheaf of OX -modules. Then there exists a −1 non-empty open subscheme U ⊂ S such that the restriction of F to XU = p (U) is flat over OU . Existence of Flattening Stratification

Theorem 4.3 Let S be a noetherian scheme, and let F be a coherent sheaf on the n projective space PS over S. Then the set I of Hilbert polynomials of restrictions of n F to fibers of PS → S is a finite set. Moreover, for each f ∈ I there exist a locally closed subscheme Sf of S, such that the following conditions are satisfied.

(i) Point-set: The underlying set |Sf | of Sf consists of all points s ∈ S where n the Hilbert polynomial of the restriction of F to Ps is f. In particular, the subsets |Sf |⊂|S| are disjoint, and their set-theoretic union is |S|. ′ (ii) Universal property: Let S = Sf be the coproduct of the Sf , and let ′ ∗ (i : S → S be the morphism induced by the` inclusions Sf ֒→ S. Then the sheaf i (F n ′ ′ on PS′ is flat over S . Moreover, i : S → S has the universal property that for any ∗ n morphism ϕ : T → S the pullback ϕ (F) on PT is flat over T if and only if ϕ factors ′ through i : S → S. The subscheme Sf is uniquely determined by the polynomial f. (iii) Closure of strata: Let the set I of Hilbert polynomials be given a total ordering, defined by putting f

19 Proof It is enough to prove the theorem for open subschemes of S which cover S, as the resulting strata will then glue together by their universal property. n Special case: Let n = 0, so that PS = S. For any s ∈ S, the fiber F|s of F over s will mean the pull-back of F to the subscheme Spec κ(s), where κ(s) is the residue field at s. (This is obtained by tensoring the stalk of F at s with the residue field at s, both regarded as OS,s-modules.) The Hilbert polynomial of the restriction of F to the fiber over s is the degree 0 polynomial e ∈ Q[λ], where e = dimκ(s) F|s.

By Nakayama lemma, any basis of F|s prolongs to a neighbourhood U of s to give a set of generators for F |U . Repeating this argument, we see that there exists a smaller neighbourhood V of s in which there is a right-exact sequence

⊕m ψ ⊕e φ OV → OV →F → 0

Let Ie ⊂ OV be the ideal sheaf formed by the entries of the e × m matrix (ψi,j) of ⊕m ψ ⊕e the homomorphism OV → OV . Let Ve be the closed subscheme of V deﬁned by Ie. For any morphism of schemes f : T → V , the pull-back sequence

∗ ∗ ⊕m f ψ ⊕e f φ ∗ OT → OT → f F → 0 is exact, by right-exactness of tensor products. Hence the pull-back f ∗F is a locally ∗ free OT -module of rank e if and only if f ψ = 0, that is, f factors via the subscheme Ve ֒→ V deﬁned by the vanishing of all entries ψi,j. Thus we have proved assertions (i) and (ii) of the theorem.

As the rank of the matrix (ψi,j) is lower semi-continuous, it follows that the function e is upper semi-continuous, which proves the assertion (iii) of the theorem, completing its proof when n = 0. General case: We now allow the integer n to be arbitrary. The idea of the proof is as follows: We show the existence of a stratification of S which is a ‘g.c.d.’ of the flattening stratifications for direct images π∗F(i) for all i ≥ N for some integer N (where the flattening stratifications for π∗F(i) exist by case n = 0 which we have treated above). This is the desired flattening stratification of F over S, as follows from Lemma 3.2. As S is noetherian, it is a finite union of irreducible components, and these are closed in S. Let Y be an irreducible component of S, and let U be the non-empty open subset of Y which consists of all points which do not lie on any other irreducible component of S. Let U be given the reduced subscheme structure. Note that this makes U an integral scheme, which is a locally closed subscheme of S. By Theorem 4.2 on generic flatness, U has a non-empty open subscheme V such that the n restriction of F to PV is flat over OV . Now repeating the argument with S replaced by its reduced closed subscheme S − V , it follows by noetherian induction on S that there exist finitely many reduced, locally closed, mutually disjoint subschemes Vi of Pn S such that set-theoretically |S| is the union of the |Vi| and the restriction of F to Vi is flat over OVi . As each Vi is a noetherian scheme, and as the Hilbert polynomials are locally constant for a flat family of sheaves, it follows that only finitely many

20 n polynomials occur in Vi in the family of Hilbert polynomials Ps(m)= χ(Ps , Fs(m)) as s varies over points of Vi. This allows us to conclude the following:

n (A) Only ﬁnitely many distinct Hilbert polynomials Ps(m)= χ(Ps , Fs(m)) occur, as s varies over all of S.

n By the semi-continuity theorem applied to the ﬂat families FVi = F|P parametrised Vi by the ﬁnitely many noetherian schemes Vi, we get the following:

r (B) There exists an integer N1 such that R π∗F(m)= 0 for all r ≥ 1 and m ≥ N1, r n and moreover H (Ps , Fs(m)) = 0 for all s ∈ S.

For each Vi, by Lemma 3.1 there exists an integer ri ≥ N1 with the property that for any m ≥ ri the base change homomorphism

(π∗F(m))|Vi → πi∗FVi (m)

Pn Pn is an isomorphism, where FVi denotes the restriction of F to Vi , and πi : Vi → Vi the projection. As the higher cohomologies of all fibers (in particular, the first cohomology) vanish by (B), it follows by semi-continuity theory for the flat family

FVi over Vi that for any s ∈ Vi the base change homomorphism

0 n (πi∗FVi (m))|s → H (Ps , Fs(m)) is an isomorphism for m ≥ ri. Taking N to be the maximum of all ri over the ﬁnitely many non-empty Vi, and composing the above two base change isomorphisms, we get the following.

0 n (π∗F(m))|s → H (Ps , Fs(m)) is an isomorphism for all m ≥ N and s ∈ S.

Note We now forget the subschemes Vi but retain the facts (A), (B), (C) which were proved using the Vi.

n Let π : PS → S denote the projection. Consider the coherent sheaves E0,...,En on S, defined by Ei = π∗F(N + i) for i =0,...,n. By applying the special case of of the theorem (where the relative dimension n of n 0 PS is 0) to the sheaf E0 on PS = S, we get a stratification (We0 ) of S indexed by ∗ integers e0, such that for any morphism f : T → S the pull-back f E0 is a locally free OT -module of rank e0 if and only if f factors via We0 ֒→ S. Next, for each stratum We0 , we take the flattening stratification (We0,e1 ) for E1|We0 , and so on. Thus in n + 1 steps, we obtain finitely many locally closed subschemes

We0,...,en ⊂ S

21 ∗ such that for any morphism f : T → S the pull-back f Ei for i =0,...,n is a locally

.free OT -modules of of constant rank ei if and only if f factors via We0,...,en ֒→ S For any integer N and n where n ≥ 0, there is a bijection from the set of numerical polynomials f ∈ Q[λ] of degree ≤ n to the set Zn+1, given by

f 7→ (e0,...,en) where ei = f(N + i).

n+1 Thus, each tuple (e0,...,en) ∈ Z can be uniquely replaced by a numerical polynomial f ∈ Q[λ] of degree ≤ n, allowing us to re-designate We0,...,en ⊂ S as Wf ⊂ S. r n Note that at any point s ∈ S, by (B) we have H (Ps , Fs(m)) = 0 for all r ≥ 1 and n m ≥ N. The polynomial Ps(m)= χ(Ps , Fs(m)) has degree ≤ n, so it is determined by its n + 1 values Ps(N),...,Ps(N + n). This shows that at any point s ∈ Wf , the Hilbert polynomial Ps(m) equals f. The desired locally closed subscheme Sf ⊂ S, whose existence is asserted by the theorem, will turn out to be a certain closed subscheme Sf ⊂ Wf whose underlying subset is all of |Wf |. The scheme structure of Sf (which may in general diﬀer from that of Wf ) is deﬁned as follows. For any i ≥ 0 and s ∈ S, the base change homomorphism

0 n (π∗F(N + i))|s → H (Ps , Fs(N + i)) is an isomorphism by statement (C). Hence each π∗F(N + i) has ﬁbers of constant rank f(N + i) on the subscheme Wf . However, this does not mean π∗F(N + i) restricts to a locally constant sheaf of rank f(N + i). But it means that Wf has a i closed subscheme Wf , whose underlying set is all of |Wf |, such that π∗F(N + i) is (i) locally free of rank f(N + i) when restricted to Wf , and moreover has the property that any base-change T → S under which π∗F(N + i) pulls back to a locally free i (i) sheaf of rank f(N + i) factors via Wf . The scheme structure of Wf is deﬁned by a coherent ideal sheaf Ii ⊂ OWf . Let I ⊂ OWf be the sum of the Ii over i ≥ 0. By noetherian condition, the increasing sequence

I0 ⊂ I0 + I1 ⊂ I0 + I1 + I2 ⊂ . . . terminates in ﬁnitely many steps, showing I is again a coherent ideal sheaf. Let Sf ⊂ Wf be the closed subscheme deﬁned by the ideal sheaf I. Note therefore that |Sf | = |Wf | and for all i ≥ 0, the sheaf π∗F(N + i) is locally free of rank f(N + i) when restricted to Sf .

It follows that from their deﬁnition that the Sf satisfy property (i) of the theorem.

We now show that the morphism f Sf → S indeed has the property (ii) of the theorem. By Lemma 3.1, there exists` some N ′ ≥ N such that for all i ≥ N ′, the base-change (π∗F(i))|Sf → (πSf )∗FSf (i) is an isomorphism for each Sf . Therefore ′ FSf is ﬂat over Sf by Lemma 3.2, as the direct images π∗F(i) for all i ≥ N are locally free over Sf . Conversely, if φ : T → S is a morphism such that FT is ﬂat, then the Hilbert polynomial is locally constant over T . Let Tf be the open and closed subscheme of T where the Hilbert polynomial is f. Clearly, the set map |Tf |→|S| factors via |Sf |. But as the direct images πT ∗FT (i) are locally free of rank f(i) on

22 Tf , it follows in fact that the schematic morphism Tf → S factors via Sf , proving the property (ii) of the theorem. As by (A) only ﬁnitely many polynomials f occur, there exists some p ≥ N such that for any two polynomials f and g that occur, we have f

Exercise What is the ﬂattening stratiﬁcation of S for the coherent sheaf OSred on S, where Sred is the underlying reduced scheme of S?

5 Construction of Quot Schemes

Notions of Projectivity Let S be a noetherian scheme. Recall that as deﬁned by Grothendieck, a morphism X → S is called a projective morphism if there exists a coherent sheaf E on P S, together with a closed embedding of X into (E) = Proj SymOS E over S. Equivalently, X → S is projective when it is proper and there exists a relatively very ample line bundle L on X over S. These conditions are related by taking L to be the restriction of OP(E)(1) to X, or in the reverse direction, taking E to be the direct image of L on S. A morphism X → S is called quasi-projective if it factors .as an open embedding X ֒→ Y followed by a projective morphism Y → S A stronger version of projectivity was introduced by Altman and Kleiman: a morphism X → S of noetherian schemes is called strongly projective (respectively, strongly quasi-projective) if there exists a vector bundle E on S together with a closed embedding (respectively, a locally closed embedding) X ⊂ P(E) over S. Finally, the strongest version of (quasi-)projectivity is as follows (used for example in the textbook [H] by Hartshorne): a morphism X → S of noetherian schemes is n projective in the strongest sense if X admits a (locally-)closed embedding into PS for some n. Note that none of the three versions of projectivity is local over the base S. Exercises (i) Gives examples to show that the above three notions of projectivity are in general distinct. (ii) Show that if X → S is projective and ﬂat, where S is noetherian, then X → S is strongly projective.

(iii) Note that if every coherent sheaf of OS-modules is the quotient of a vector bundle, then projectivity over the base S is equivalent to strong projectivity. If S admits an ample line bundle (for example, if S is quasi-projective over an aﬃne base), then all three notions of projectivity over S are equivalent to each other.

23 Main Existence Theorems Grothendieck’s original theorem on Quot schemes, whose proof is outlined in [FGA] TDTE-IV, is the following.

Theorem 5.1 (Grothendieck) Let S be a noetherian scheme, π : X → S a projective morphism, and L a relatively very ample line bundle on X. Then for any Φ,L coherent OX -module E and any polynomial Φ ∈ Q[λ], the functor QuotE/X/S is Φ,L representable by a projective S-scheme QuotE/X/S.

Altman and Kleiman gave a complete and detailed proof of the existence of Quot schemes in [A-K 2]. They could remove the noetherian hypothesis, by instead as- suming strong (quasi-)projectivity of X → S together with an assumption about Φ,L the nature of the coherent sheaf E, and deduce that the scheme QuotE/X/S is then strongly (quasi-)projective over S. For simplicity, in these lecture notes we state and prove the result in [A-K 2] in the noetherian context.

Theorem 5.2 (Altman-Kleiman) Let S be a noetherian scheme, X a closed subscheme of P(V ) for some vector bundle V on S, L = OP(V )(1)|X, E a coherent quotient sheaf of π∗(W )(ν) where W is a vector bundle on S and ν is an integer, Φ,L Φ,L and Φ ∈ Q[λ]. Then the functor QuotE/X/S is representable by a scheme QuotE/X/S which can be embedded over S as a closed subscheme of P(F ) for some vector bundle F on S. The vector bundle F can be chosen to be an exterior power of the tensor product of W with a symmetric powers of V .

Taking both V and W to be trivial in the above, we get the following.

n Theorem 5.3 If S is a noetherian scheme, X is a closed subscheme of PS for some n p n ≥ 0, L = OPS (1)|X, E is a coherent quotient sheaf of ⊕ OX (ν) for some integers Φ,L p ≥ 0 and ν, and Φ ∈ Q[λ], then the the functor QuotE/X/S is representable by a Φ,L r scheme QuotE/X/S which can be embedded over S as a closed subscheme of PS for some r ≥ 0.

The rest of this section is devoted to proving Theorem 5.2, with extra noetherian hypothesis. At the end, we will remark on how the proof also gives us the original version of Grothendieck. Φ,L Reduction to the case of Quotπ∗W/P(V )/S It is enough to prove Theorem 5.2 in the special case that X = P(V ) and E = π∗(W ) where V and W are vector bundles on S, as a consequence of the next lemma.

24 Lemma 5.4 (i) Let ν be any integer. Then tensoring by Lν gives an isomorphism of Φ,L Ψ,L functors from QuotE/X/S to QuotE(ν)/X/S where the polynomial Ψ ∈ Q[λ] is deﬁned by Ψ(λ)=Φ(λ + ν). (ii) Let φ : E → G be a surjective homomorphism of coherent sheaves on X. Then Φ,L Φ,L the corresponding natural transformation QuotG/X/S → QuotE/X/S is a closed embedding.

Proof The statement (i) is obvious. The statement (ii) just says that given any Φ,L locally noetherian scheme T and a family hF, qi ∈ QuotE/X/S(T ), there exists a closed subscheme T ′ ⊂ T with the following universal property: for any locally noetherian scheme U and a morphism f : U → T , the pulled back homomorphism of OXU -modules qU : EU → FU factors via the pulled back homomorphism φU : ′ ′ EU → GU if and only if U → T factors via T ֒→ T . This is satisﬁed by taking T q to be the vanishing scheme for the composite homomorphism ker(φ) ֒→ E → F of coherent sheaves on XT (see Remark 3.6), which makes sense here as both ker(φ) and F are coherent on XT and F is ﬂat over T .

Φ,L Therefore if Quotπ∗W/P(V )/S is representable, then for any coherent quotient E of ∗ Φ,L Φ,L π W (ν)|X , we can take QuotE/X/S to be a closed subscheme of Quotπ∗W/P(V )/S . Use of m-Regularity We consider the sheaf E = π∗(W ) on X = P(V ) where V is a vector bundle on S, and take L = OP(V )(1). For any ﬁeld k and a k-valued point s of S, we have an n isomorphism P(V )s ≃ Pk where n = rank(V ) − 1, and the restricted sheaf Es on p P(V )s is isomorphic to ⊕ OP(V )s where p = rank(W ). It follows from Theorem 2.3 that given any Φ ∈ Q[λ], there exists an integer m which depends only on rank(V ), rank(W ) and Φ, such that for any ﬁeld k and a k-valued point s of S, the sheaf Es on P(V )s is m-regular, and for any coherent quotient q : Es → F on P(V )s with Hilbert polynomial Φ, the sheaf F and the kernel sheaf G ⊂ Es of q are both m-regular. In particular, it follows from the Castelnuovo Lemma 2.1 that for r ≥ m, i i i all cohomologies H (Xs, Es(r)), H (Xs, F(r)), and H (Xs, G(r)) are zero for i ≥ 1, 0 0 0 and H (Xs, Es(r)), H (Xs, F(r)), and H (Xs, G(r)) are generated by their global sections.

From the above it follows by Theorem 3.7 that if T is an S-scheme and q : ET →F is a T -ﬂat coherent quotient with Hilbert polynomial Φ, then we have the following, where G ⊂ ET is the kernel of q.

(*) The sheaves πT ∗G(r), πT ∗ET (r), πT ∗F(r) are locally free of ﬁxed ranks de- ∗ termined by the data n, p, r, and Φ, the homomorphisms πT πT ∗(G(r)) → G(r), ∗ ∗ πT πT ∗(ET (r)) → ET (r), πT πT ∗(F(r)) →F(r) are surjective, and the higher direct i i i images R πT ∗G(r), R πT ∗ET (r), R πT ∗F(r) are zero, for all r ≥ m and i ≥ 1. (**) In particular we have the following commutative diagram of locally sheaves on XT , in which both rows are exact, and all three vertical maps are surjective.

25 ∗ ∗ ∗ 0 → πT πT ∗(G(r)) → πT πT ∗(ET (r)) → πT πT ∗(F(r)) → 0 ↓ ↓ ↓ 0 → G(r) → E(r) → F(r) → 0

Embedding Quot into Grassmannian

We now ﬁx a positive integer r such that r ≥ m. Note that the rank of πT ∗F(r) r is Φ(r) and π∗E(r) = W ⊗OS Sym V . Therefore the surjective homomorphism r πT ∗ET (r) → πT ∗F(r) deﬁnes an element of the set Grass(W ⊗OS Sym V, Φ(r))(T ). We thus get a morphism of functors

Φ,L r α : QuotE/X/S → Grass(W ⊗OS Sym V, Φ(r))

It associates to q : ET →F the quotient πT ∗(q(r)) : πT ∗ET (r) → πT ∗F(r).

The above morphism α is injective because the quotient q : ET →F can be recovered from πT ∗(q(r)) : πT ∗ET (r) → πT ∗F(r) as follows. r ∗ If G = Grass(W ⊗OS Sym V, Φ(r)) with projection pG : G → S, and u : pG E → U ∗ denotes the universal quotient on G with kernel v : K → pG E, then the ho- ∗ ∗ momorphism πT πT ∗(G(r)) → πT πT ∗ET (r) can be recovered from the morphism ∗ ∗ T → G as the pull-back of v : K→ pG E. Let h be the composite πT πT ∗(G(r)) → ∗ πT πT ∗(ET (r)) → ET (r). As a consequence of the properties of the diagram (**), the following is a right exact sequence on XT

∗ h q(r) πT πT ∗(G(r)) → ET (r) →F→ 0 and so q(r) : ET (r) →F(r) can be recovered as the cokernel of h. Finally, twisting by −r, we recover q, proving the desired injectivity of the morphism of functors Φ,L r α : QuotE/X/S → Grass(W ⊗OS Sym V, Φ(r)). Use of Flattening Stratiﬁcation

Φ,L r We will next prove that α : QuotE/X/S → Grass(W ⊗OS Sym V, Φ(r)) is relatively representable. In fact, we will show that given any locally noetherian S-scheme T r and a surjective homomorphism f : WT ⊗OT Sym VT →J where J is a locally free ′ OT -module of rank Φ(r), there exists a locally closed subscheme T of T with the following universal property (F) : (F) Given any locally noetherian S-scheme Y and an S-morphism φ : Y → T , let ∗ fY be the pull-back of f, and let KY = ker(fY ) = φ ker(f). Let πY : XY → Y be ∗ the projection, and let h : πY KY → EY be the composite map

∗ ∗ r ∗ πY KY → πY (W ⊗OS Sym V )= πY πY ∗EY → EY

Let q : EY → F be the cokernel of h. Then F is ﬂat over Y with its Hilbert . polynomial on all ﬁbers equal to Φ if and only if φ : Y → T factors via Y ′ ֒→ Y

26 The existence of such a locally closed subscheme T ′ of T is given by Theorem 4.3, which shows that T ′ is the stratum corresponding to Hilbert polynomial Φ for the flattening stratification over T for the sheaf F on XT . r When we take T to be Grass(W ⊗OS Sym V, Φ(r)) with universal quotient u : ∗ ′ pG E → U, the corresponding locally closed subscheme T represents the functor Φ,L QuotE/X/S by its construction. Φ,L Hence we have shown that QuotE/X/S is represented by a locally closed subscheme r r of Grass(W ⊗OS Sym V, Φ(r)). As Grass(W ⊗OS Sym V, Φ(r)) embeds as a closed Φ(r) r subscheme of P( W ⊗OS Sym V ), we get a locally closed embedding of S- schemes V Φ(r) Φ,L r QuotE/X/S ⊂ P( (W ⊗OS Sym V )) ^ Φ,L In particular, the morphism QuotE/X/S → S is separated and of finite type. Valuative Criterion for Properness The original reference for the following argument is EGA IV (2) 2.8.1. Φ,L The functor QuotE/X/S satisfies the following valuative criterion for properness over S: given any discrete valuation ring R over S with quotient field K, the restriction map Φ,L Φ,L QuotE/X/S(Spec R) → QuotE/X/S (Spec K) is bijective. This can be seen as follows. Given any coherent quotient q : EK → F Φ,L on XR which defines an element hF, qi of QuotE/X/S(Spec K). Let F be the image of the composite homomorphism ER → j∗(EK ) → j∗F where j : XK ֒→ XR is the open inclusion. Let q : ER → F be the induced surjection. Then we leave it to Φ,L the reader to verify that hF, qi is an element of QuotE/X/S (Spec R) which maps to hF, qi, and is the unique such element. (Use the basic fact that being flat over a dvr is the same as being torsion-free.) Φ,L As S is noetherian and as we have already shown that QuotE/X/S → S is of finite Φ,L type, it follows that QuotE/X/S → S is a proper morphism. Therefore the embedding Φ,L Φ(r) r of QuotE/X/S into P( (W ⊗OS Sym V )) is a closed embedding. This completes the proofV of Theorem 5.2. The Version of Grothendieck We now describe how to get Theorem 5.1 from the above proof. As S is noetherian, we can find a common m such that given any field-valued point s : Spec k → S and a coherent quotient q : Es →F on Xs with Hilbert polynomial Φ, the sheaves Es(r), F(r), G(r) (where G = ker(q)) are generated by global sections and all their higher cohomologies vanish, whenever r ≥ m. This follows from the theory of m-regularity, and semi-continuity. Because we have such a common m, we get as before an injective morphism from Φ,L the functor QuotE/X/S into the Grassmannian functor Grass(π∗E(r), Φ(r)). The

27 sheaf π∗E(r) is coherent, but need not be the quotient of a vector bundle on S. Consequently, the scheme Grass(π∗E(r), Φ(r)) is projective over the base, but not necessarily strongly projective. Finally, the use of flattening stratification, which can be made over an affine open cover of S, gives a locally closed subscheme of Grass(π∗E(r), Φ(r)) which represents Φ,L QuotE/X/S, which is in fact a closed subscheme by the valuative criterion. Thus, we Φ,L get QuotE/X/S as a projective scheme over S.

6 Some Variants and Applications

Quot Scheme in Quasi-Projective case Exercise Let π : Z → S be a proper morphism of noetherian schemes. Let Y ⊂ Z be a closed subscheme, and let F be a coherent sheaf on Z. Then there exists an open subscheme S′ ⊂ S with the universal property that a morphism T → S factors ′ through S if and only if the support of the pull-back FT on ZT = Z ×S T is disjoint from YT = Y ×S T . Exercise As a consequence of the above, show the following: If π : Z → S is a proper morphism with S noetherian, if X ⊂ Z is an open subscheme, and if E is a coherent sheaf on Z, then QuotE|X /X/S is an open subfunctor of QuotE/Z/S. With the above preparation, the construction of a quot scheme extends to the strongly quasi-projective case, to give the following.

Theorem 6.1 (Altman and Kleiman) Let S be a noetherian scheme, X a locally closed subscheme of P(V ) for some vector bundle V on S, L = OP(V )(1)|X , E a ∗ coherent quotient sheaf of π (W )(ν)|X where W is a vector bundle on S and ν is an Φ,L integer, and Φ ∈ Q[λ]. Then the functor QuotE/X/S is representable by a scheme Φ,L QuotE/X/S which can be embedded over S as a locally closed subscheme of P(F ) for some vector bundle F on S. Moreover, the vector bundle F can be chosen to be an exterior power of the tensor product of W with a symmetric power of V .

Proof Let X ⊂ P(V ) be the schematic closure of X ⊂ P(V ), and let E be the coherent sheaf on X deﬁned as the image of the composite homomorphism

∗ ∗ π (W )(ν)|X → j∗(π (W )(ν)|X ) → j∗E

∗ Then we get a quotient π (W )(ν)|X → E which restricts on X ⊂ X to the given ∗ quotient π (W )(ν)|X → E. Therefore by the above exercise, QuotE/X/S is an open subfunctor of QuotE/X/S. Now the result follows from the Theorem 5.2.

In order to extend Grothendieck’s construction of a quot scheme to the quasi- projective case, one ﬁrst needs the following lemma which is of independent interest.

28 Lemma 6.2 Any coherent sheaf on an open subscheme of a noetherian scheme S can be prolonged to a coherent sheaf on all of S.

Proof First consider the case where S = Spec A is affine, and let j : U ֒→ S denote the inclusion. The quasi-coherent sheaf j∗(F) corresponds to the A-module 0 ∼ M = H (S, j∗(F)), in the sense that j∗(F) = M . Given any u ∈ U, there exist finitely many elements e1,...,en ∈ M which generate the fiber Fu regarded as a vector space over the residue field κ(u). By Nakayama these elements will generate the stalks of F in an open neighbourhood of u in U. Therefore by the noetherian hypothesis, there exist finitely many elements e1,...,er ∈ M which generate the stalk of F at each point of U. If N ⊂ M is the submodule generated by these elements, then G = N ∼ is a coherent prolongation of F to S = Spec A, proving the result in the affine case. In the general case, by the noetherian condition there exists a maximal coherent prolongation (U ′, F ′) of F. Then unless U ′ = S, we can obtain a further prolongation of F ′ by using the affine case. For, if u ∈ S−U ′, we can take an affine open subscheme ′ ′ V containing u, and a coherent prolongation G of F |U ′ V to all of V , and then glue together G′ and F ′ along U ′ V to further prolong FT′ to U ′ V , contradicting the maximality of (U ′, F ′). T S

Theorem 6.3 (Grothendieck) Let S be a noetherian scheme, X a quasi-projective scheme over S, L a line bundle on X which is relatively very ample over S, E a Φ,L quotient sheaf on X, and Φ ∈ Q[λ]. Then the functor QuotE/X/S is representable by Φ,L a scheme QuotE/X/S which is quasi-projective over S.

Proof By deﬁnition of quasi-projectivity of X → S, note that X can be embedded over S as a locally closed subscheme of P(V ) for some coherent sheaf V on S, such that L is isomorphic to OP(V )(1)|X . Let X ⊂ P(V ) be the schematic closure of X in P(V ). This is a projective scheme over S, and X is embedded as an open subscheme in it. By Lemma 6.2 the coherent sheaf E has a coherent prolongation E to X. For any such prolongation E, the functor QuotE/X/S is an open subfunctor of QuotE/X/S. Therefore the desired result now follows from Theorem 5.1.

Scheme of Morphisms We recall the following basic facts about ﬂatness.

Lemma 6.4 (1) Any finite-type flat morphism between noetherian schemes is open. (2) Let π : Y → X be a finite-type morphism of noetherian schemes. Then all y ∈ Y such that π is flat at y (that is, OY,y is a flat OX,π(y)-module) form an open subset of Y .

29 (3) Let S be a noetherian scheme, and let f : X → S and g : Y → S be finite type flat morphisms. Let π : Y → X be any morphism such that g = f ◦π. Let y ∈ Y , let x = π(y), and let s = g(y)= f(x). If the restricted morphism πs : Ys → Xs between the fibers over s is flat at y ∈ Ys, then π is flat at y ∈ Y .

Proof See for example Altman and Kleiman [A-K 1] Chapter V. The statement (3) is a consequence of what is known as the local criterion for ﬂatness.

Theorem 6.5 Let S be a noetherian scheme, and let f : X → S and g : Y → S be proper ﬂat morphisms. Let π : Y → X be any projective morphism with g = f ◦ π. Then S has open subschemes S2 ⊂ S1 ⊂ S with the following universal properties:

(a) For any locally noetherian S-scheme T , the base change πT : YT → XT is a ﬂat morphism if and only if the structure morphism T → S factors via S1. (This does not need π to be projective.)

(b) For any locally noetherian S-scheme T , the base change πT : YT → XT is an isomorphism if and only if the structure morphism T → S factors via S2.

Proof (a) By Lemma 6.4.(2), all y ∈ Y such that π is flat at y form an open ′ ′ subset Y ⊂ Y . Then S1 = S − g(Y − Y ) is an open subset of S as g is proper. We give S1 the open subscheme structure induced from S. It follows from the local criterion of flatness (Lemma 6.4.(3)) that S1 exactly consists of all s ∈ S such that the restricted morphism πs : Ys → Xs between the fibers over s is flat. Therefore again by the local criterion of flatness, S1 has the desired universal property. b) Let π1 : Y1 → X1 be the pull-back of π under the inclusion S1 ֒→ S. Let L be) a relatively very ample line bundle for the projective morphism π1 : Y1 → X1. Then m by noetherianness there exists an integer m ≥ 1 such that π1∗L is generated by i m its global sections and R π1∗L = 0 for all i ≥ 1. By flatness of π1, it follows that m m π1∗L is a locally free sheaf. Let U ⊂ X1 be the open subschemes such that π1∗L is of rank 1 on U. Finally, let S2 = S1 −f(X1 −U), which is open as f is proper. We give S2 the induced open subscheme structure, and leave it to the reader to verify that it indeed has the required universal property (b) .

If X and Y are schemes over a base S, then for any S-scheme T , an S-morphism from X to Y parametrised by T will mean a T -morphism from X ×S T to Y ×S T . The set of all such will be denoted by MorS(X,Y )(T ). The association T 7→ MorS(X,Y )(T ) deﬁnes a contravariant functor MorS(X,Y ) from S-schemes to Sets. Exercise Let k be a ﬁeld, let S = Spec k[[t]], X = Spec k = Spec(k[[t]]/(t)), and 1 let Y = PS. Is MorS(X,Y ) representable?

Theorem 6.6 Let S be a noetherian scheme, let X be a projective scheme over S, and let Y be quasi-projective scheme over S. Assume moreover that X is ﬂat over S.

30 Then the functor MorS(X,Y ) is representable by an open subscheme MorS(X,Y ) of HilbX×SY/S.

Proof We can associate to each morphism f : XT → YT (where T is a scheme over S) its graph ΓT (f) ⊂ (X ×S Y )T , which is closed in (X ×S Y )T by separatedness of Y → S. We regard ΓT (f) as a closed subscheme of (X ×S Y )T which is isomorphic to X under the graph morphism (idX , f) : X → ΓT (f) and the projection ΓT (f) → X, which are inverses to each other. As X is proper and ﬂat over S, so is ΓT (f), therefore this deﬁnes a set-map ΓT : MorS(X,Y )(T ) → HilbX×S Y/S(T ) which is functorial in T , so we obtain a morphism of functors

Γ : MorS(X,Y ) → HilbX×SY/S

Given any element of HilbX×S Y/S(T ), represented by a family Z ⊂ (X ×S Y )T , it follows by applying Theorem 6.5.(b) to the projection Z → X that T has an open subscheme T ′ with the following universal property: for any base-change U → T , the pull-back ZU ⊂ (X ×S Y )U maps isomorphically on to XU under the projection ′ p :(X ×S Y )U → XU if and only if U → T factors via T . Note therefore that over ′ T , the scheme ZT ′ will be the graph of a uniquely determined morphism XT ′ → YT ′ .

This shows that the morphism of functors Γ : MorS(X,Y ) → HilbX×SY/S is a representable morphism which is an open embedding. Therefore a representing scheme MorS(X,Y ) for MorS(X,Y ) exists as an open subscheme of HilbX×S Y/S.

Exercise Let S be a noetherian scheme and X → S a ﬂat projective morphism. Consider the set-valued contravariant functor AutX/S on locally noetherian S-schemes, which associates to any T the set of all automorphisms of XT over T . Show that this functor is representable by an open subscheme of MorS(X,X). Exercise Let S be a noetherian scheme and π : Z → X a morphism of S-schemes, where X is proper over S and Z is quasi-projective over S. Consider the set-valued contravariant functor ΠZ/X/S on locally noetherian S-schemes, which associates to any T the set of all sections of πT : ZT → XT . Show that this functor is representable by an open subscheme of HilbZ/S.

Quotient by a Flat Projective Equivalence Relation Let X be a scheme over a base S. A schematic equivalence relation on X over S will mean an S-scheme R together with a morphism f : R → X ×S X over S such that for any S-scheme T the set map f(T ) : R(T ) → X(T ) × X(T ) is injective and its image is the graph of an equivalence relation on X(T ). (Here, we denote by Z(T ) the set MorS (T,Z) = hZ (T ) of all T -valued points of Z, where Z and T are S-schemes.) We will say that a morphism q : X → Q of S-schemes is a quotient for a schematic equivalence relation f : R → X ×S X over S if q is a co-equaliser for the component → morphisms f1, f2 : R → X of f : R → X ×S X. This means q ◦ f1 = q ◦ f2, and given

31 any S-scheme Z and an S-morphism g : X → Z such that g ◦ f1 = g ◦ f2, there exists a unique S-morphism h : Q → Z such that g = h ◦ q. A schematic quotient q : X → Q, when it exists, is unique up to a unique isomorphism. Exercise: A schematic quotient, when it exists, is necessarily an epimorphism in the category of S-schemes. Caution Even if q : X → Q is a schematic quotient for R, for a given T the map q(T ) : X(T ) → Q(T ) may not be a quotient for R(T ) in the category of sets. The map q(T ) may fail to be surjective, and moreover it may identify two distinct equivalence classes. Exercise: Give examples where such phenomena occur. We will say that the quotient q : X → Q is eﬀective if the induced morphism (f1, f2) : R → X ×Q X is an isomorphism of S-schemes. In particular, it will ensure that distinct equivalence classes do not get identiﬁed under q(T ) : X(T ) → Q(T ). But q(T ) can still fail to be surjective, as in the following example.

n Exercise Let S = Spec Z, and let X ⊂ AZ be the complement of the zero section n n of AZ. Note that for any ring B, an element of X(Spec B) is a vector u ∈ B such that at least one component of u is invertible in B. Show that X ×S X has a closed subscheme R whose B-valued points for any ring B are all pairs (u, v) ∈ X(Spec B) × X(Spec B) such that there exists an invertible element λ ∈ B× with n−1 λu = v. Show that an effective quotient q : X → Q exists, where Q = PZ . However, show that q does not admit a global section, and so q(Q) : X(Q) → Q(Q) is not surjective. The famous example by Hironaka (see Example 3.4.1 in Hartshorne [H] Appendix B) of a non-projective smooth complete variety X over C together with a schematic equivalence relation R (for which the morphisms fi : R → X are finite flat, in fact, étale of degree 2) shows that schematic quotients do not always exist. But under the powerful assumption of projectivity, Grothendieck proved an existence result for quotients, to which we devote the rest of this section. We will need the following elementary lemma from Grothendieck’s theory of faithfully flat descent (this is a special case of [SGA 1] ExposéVIII Corollary 1.9). The reader can consult the lectures of Vistoli [V] for an exposition of descent.

Lemma 6.7 (1) Any faithfully flat quasi-compact morphism of schemes f : X → Y is an effective epimorphism, that is, f is a co-equaliser for the projections p1,p2 : → X ×Y X → X. (2) Let p : D → H be a faithfully flat quasi-compact morphism. Let Z ⊂ D be a closed subscheme such that

−1 −1 p1 Z = p2 Z ⊂ D ×H D

→ −1 where p1,p2 : D ×H D → D are the projections, and pi Z is the schematic inverse image of Z under pi. Then there exists a unique closed subscheme Q of H such that Z = p−1Q ⊂ D. By base-change from p : D → H, it follows that the induced morphism p|Z : Z → Q is faithfully ﬂat and quasi-compact.

32 The idea of using Hilbert schemes to make quotients of flat projective equivalence relations is due to Grothendieck, who used it in his construction of a relative Picard scheme. In set-theoretic terms, the idea is actually very simple: Let X be a set, and R ⊂ X × X an equivalence relation on X. Let H be the power set of X (means the set of all subsets of X), and let ϕ : X → H be the map which sends x ∈ X to its equivalence class [x] ∈ H. If Q ⊂ H is the image of ϕ, then the induced map q : X → Q is the quotient of X modulo R in the category of sets. The scheme- theoretic analogue of the above is the following theorem of Grothendieck, where the Hilbert scheme of X plays the role of power set. The first detailed proof appeared in Altman and Kleiman [A-K 2]. Theorem 6.8 Let S be a noetherian scheme, and let X → S be a quasi-projective morphism. Let f : R → X ×S X be a schematic equivalence relation on X over → S, such that the projections f1, f2 : R → X are proper and flat. Then a schematic quotient X → Q exists over S. Moreover, Q is quasi-projective over S, the morphism X → Q is faithfully flat and projective, and the induced morphism (f1, f2) : R → X ×Q X is an isomorphism (the quotient is effective).

Proof (Following Altman and Kleiman [A-K 2]) The properness of fi together with separatedness of X → S implies properness of f : R → X ×S X. Also, f is functorially injective by definition of a schematic equivalence relation. It follows that f is a closed embedding, which allows us to regard R as a closed subscheme of X ×S X (Exercise: Any proper morphism of noetherian schemes, which is injective at the level of functor of points, is a closed embedding). This defines an element (R) of HilbX/S(X), as the projection p2|R = f2 is proper and flat.

By Theorem 6.3, there exists a scheme HilbX/S which represents the functor HilbX/S. As the parameter scheme X is noetherian and as the Hilbert polynomial is locally constant, only finitely many polynomials Φ occur as Hilbert polynomials of fibers of f2 : R → X with respect to a chosen relatively very ample line bundle L on X over S. Let H be the finite disjoint union of the corresponding open subschemes Φ,L Φ,L HilbX/S of HilbX/S. Then H is a quasi-projective scheme over S as each HilbX/S is so by Theorem 6.3. The family (R) ∈ HilbX/S (X) therefore defines a classifying ∗ morphism ϕ : X → H, with the property that ϕ D = R where D ⊂ X ×S H ∗ denotes the restriction to H of the universal family over HilbX/S, and ϕ D denotes −1 (idX × ϕ) D. Also, note that the projection p : D → H is proper and flat. If X is non-empty then each fiber of f2 : R → X is also non-empty as the diagonal ∆X is contained in R, and therefore the Hilbert polynomial of each fiber is non-zero. Hence p : D → H is surjective, and so p is faithfully flat. For any S-scheme T , it follows from its definition that a T -valued point of D is a pair (x, V ) with x ∈ X(T ) and V ∈ H(T ) such that x ∈ V where the notation “x ∈ V ” more precisely means that the graph morphism (x, idT ) : T → X ×S T factors via V ⊂ X ×S T . With this notation, we will establish the following crucial property:

33 (***) For any S-scheme T , and T -valued points x, y ∈ X(T ), the following equiva- lences hold: (x, y) ∈ R(T ) ⇔ x ∈ ϕ(y) ⇔ ϕ(x)= ϕ(y) ∈ H(T ).

For this, note that for any x, y ∈ X(T ), the morphism (x, y) : T → X ×S X factors as the composite (x,idT ) idX × y T → X ×S T → X ×S X As ϕ∗D = R and (ϕ ◦ y)∗D = ϕ(y), it follows that y∗R = ϕ(y), in other words, the schematic inverse image of R ⊂ X ×S X under idX × y : X ×S T → X ×S X is ϕ(y) ⊂ X ×S T . Hence the above factorisation of (x, y) : T → X ×S X shows that

(x, y) ∈ R(T )

⇔ the morphism (x, idT ) : T → X ×S T factors via ϕ(y) ⊂ X ×S T ⇔ x ∈ ϕ(y)

Moreover, x ∈ ϕ(x) as ∆X ⊂ R. Therefore, if ϕ(x)= ϕ(y) then x ∈ ϕ(y). It now only remains to prove that if (x, y) ∈ R(T ) then ϕ(x) = ϕ(y), that is, the ∗ subschemes ϕ(x) and ϕ(y) of X ×S T are identical. Note that ϕ(x)=(ϕ ◦ x) D = x∗ϕ∗D = x∗R and similarly ϕ(y)= y∗R, therefore we wish to show that x∗R = y∗R. To show this in terms of functor of points, for any T -scheme u : U → T we just have ∗ ∗ ∗ to show that (x R)(U)=(y R)(U) as subsets of (X ×S T )(U). As x R is the inverse image of R under idX × x : X ×S T → X ×S X, it follows that a U-valued point of ∗ x R is the same as an element z ∈ X(U) such that (idX × x) ◦ (z,u) ∈ R(U). But as (idX × x) ◦ (z,u)=(z, x ◦ u), it follows that

z ∈ (x∗R)(U) ⇔ (z, x ◦ u) ∈ R(U)

As R(U) is an equivalence relation on the set X(U), and as by assumption (x, y) ∈ R(T ), we have (x ◦ u,y ◦ u) ∈ R(U), and so by transitivity we have

z ∈ (x∗R)(U) ⇔ (z, x ◦ u) ∈ R(U) ⇔ (z,y ◦ u) ∈ R(U) ⇔ z ∈ (y∗R)(U)

∗ ∗ Hence the subschemes x R and y R of X ×S T have the same U-valued points for any T -scheme U, and therefore x∗R = y∗R, as was to be shown. This completes the proof of the assertion (***).

The graph morphism (idX ,ϕ) : X → X×S H is a closed embedding as H is separated ∗ over S. As ∆X ⊂ R and as ϕ D = R, it follows that (idX ,ϕ) factors through D ⊂ X ×S H. Thus, we get a closed subscheme Γϕ ⊂ D, which is the isomorphic image of X under (idX ,ϕ). We wish to apply the Lemma 6.7.(2) to the faithfully ﬂat quasi-compact morphism p : D → H and the closed subscheme Z =Γϕ ⊂ D.

Any T -valued point of Γϕ is a pair (x, ϕ(x)) ∈ D(T ) where x ∈ X(T ). Any T -valued point of D ×H D is a triple (x,y,V ) where x, y ∈ X(T ) and V ∈ H(T ) such that → x, y ∈ V . Under the projections p1,p2 : D ×H D → D, we have p1(x,y,V )=(x, V ) and p2(x,y,V )=(y,V ). We now have

p1(x,y,V ) ∈ Γϕ(T )

34 ⇔ (x, V ) ∈ Γϕ(T ) and y ∈ V ⇔ V = ϕ(x) and y ∈ V ⇔ y ∈ ϕ(x)= V ⇔ ϕ(y)= ϕ(x)= V (by the property (***)).

Similarly, we have p2(x,y,V ) ∈ Γϕ(T ) if and only if ϕ(y) = ϕ(x) = V . Therefore, p1(x,y,V ) ∈ Γϕ(T ) if and only if p2(x,y,V ) ∈ Γϕ(T ). This holds for all T -valued −1 −1 points for all S-schemes T , and so p1 Γϕ = p2 Γϕ ⊂ D ×H D. Therefore by Lemma 6.7(2) there exists a unique closed subscheme Q ⊂ H such that Γϕ is the pull-back of Q under D → H. Let p :Γϕ → Q be the morphism induced by the restriction to Γϕ of p : D → H. Let q : X → Q be deﬁned as the composite

(idX ,ϕ) p X → Γϕ → Q

q .Then note that the composite X → Q ֒→ H equals ϕ We will now show that q : X → Q as defined above is the desired quotient of X by R, with the required properties. (i) Quasi-projectivity of Q → S : This is satisfied as Q is closed in H and H is quasi-projective over S. (ii) Faithful flatness and projectivity of q : This follows by base change from the faithfully flat projective morphism p : D → H, as the following square is Cartesian.

(id ,ϕ) X →X D q ↓ ↓ p Q ֒→ H

→ (iii) Exactness of R → X → Q and the isomorphism R → X ×Q X : By (***), for any T -valued points x, y ∈ X(T ), we have (x, y) ∈ R(T ) if and only if ϕ(x) = f1 q f2 q ϕ(y). This shows that the composite R → X → Q equals the composite R → X → Q, and the induced morphism (f1, f2) : R → X ×Q X is an isomorphism, by showing these statements hold at the level of functor of points. Under the isomorphism → R → X ×Q X, the morphisms f1, f2 : R → X become the projection morphisms → p1,p2 : X ×Q X → X. By Lemma 6.7, the morphism q : X → Q is a co-equaliser for p1,p2, and so q is a co-equaliser for f1, f2. This completes the proof of Theorem 6.8.

What Altman and Kleiman actually prove in [A-K 2] is a strongly projective form of the above theorem (without a noetherian assumption), using the hypothesis of strong quasi-projectivity in the following places in the above proof: if X → S is strongly quasi-projective then H → S will again be so by Theorem 6.1, and therefore Q will be strongly quasi-projective over S. Moreover, D → H will be strongly projective, and therefore by base-change X → Q will be strongly projective. In the noetherian case, this gives us the following result.

35 Theorem 6.9 Let S be a noetherian scheme, and let X → S be a strongly quasi- projective morphism. Let f : R → X ×S X be a schematic equivalence relation → on X over S, such that the projections f1, f2 : R → X are proper and flat. Then a schematic quotient X → Q exists over S. Moreover, the quotient is effective, the morphism X → Q is faithfully flat and strongly projective, and Q is strongly quasi-projective over S.

References

[EGA] Grothendieck, A. : Eléments´ de Géométrie Algébriques (written with the collaboration of Dieudonné, J.). Publ. Math. IHES, volumes 4, 8, 11, 17, 20, 24, 28, 32 (1960-67). [FGA] Grothendieck, A. : Fondements de la Géométrie Algébriques. (Collection of six lectures in Séminaire Bourbaki, delivered during 1957-1962, including [TDTE IV]). Secrétariat mathématique, 11 rue Pierre Curie, Paris 5e, 1962. [TDTE IV] Grothendieck, A. : Techniques de construction et théorèmes d’existence en géométrie algébriques IV : les schémas de Hilbert. Séminaire Bourbaki 221, 1960/61. [SGA 1] Grothendieck, A. : Revtements tales et groupe fondamental, Springer LNM 224, 1971. This is vloume 1 of Séminaire de la Géométrie Algébriques. Large parts of SGA are now available on-line, as scans or as text, through efforts of many mathematicians. [A-K 1] Altman, A. and Kleiman, S. : Introduction to Grothendieck duality theory, Springer LNM 146, 1970. [A-K 2] Altman, A. and Kleiman, S. : Compactifying the Picard scheme. Advances in Math. 35 (1980) 50-112. [B-L-R] Bosch, S., Lütkebohmert, W., Raynaud, M. : Néron Models, Springer 1990. [H] Hartshorne, R. : Algebraic Geometry. Springer GTM 52, 1977. [K] Kleiman, S. : An intersection theory for divisors. (Preprint, 1994). [M] Mumford, D. : Lectures on Curves on an Algebraic Surface. Princeton Uni- versity Press, 1966. [V] Vistoli, A. : Notes on Grothendieck topologies, fibered categories, and descent theory. Based on lectures in the Summer School ‘Advanced Basic Algebraic Geometry’, Abdus Salam ICTP July 2003. e-print: arXiv.org/math.AG/0412512